Knot Symmetries and the Fundamental Quandle

KNOT SYMMETRIES AND THE FUNDAMENTAL QUANDLE

EVA HORVAT

Abstract. We establish a relationship between the knot symmetries and the automorphisms of the knot quandle. We identify the homeomorphisms of the pair (S3,K) that induce the (anti)automorphisms of the fundamental quandle Q(K). We show that every quandle (anti)automorphism of Q(K) is induced by a homeomorphism of the pair (S3,K). As an application of those results, we are able to explore some symmetry properties of a knot based on the presentation of its fundamental quandle, which is easily derived from a knot diagram.

1. Introduction It is well known that the knot quandle is a complete knot invariant [8]. The knot quandle and various derived quandle invariants have been extensively used to study and distinguish nonequivalent knots. Our idea is to study knot equivalences from the viewpoint of the fundamental quandle. In this paper, we establish a relationship between the knot symmetries and the automorphisms of the knot quandle. Our goal is to investigate the complex topological information, hidden in the group of knot symmetries, from a purely algebraical perspective of the knot quandle. Our main results are the following.

3 3 Proposition 1.1. Let f :(S ,K) → (S ,K) be a homeomorphism for which [f|∂NK ] = ±1 ∈ 2 MCG(T ). Then f induces a map f∗ : Q(K) → Q(K) that is either a quandle automorphism or a quandle antiautomorphism. Proposition 1.2. Let F : Q(K) → Q(K) be an (anti)automorphism of the fundamental quandle of a nontrivial knot K. Then F is induced by a homeomorphism that preserves the orientation of the normal bundle if F is an automorphism, and reverses the orientation of the normal bundle if F is an antiautomorphism.

arXiv:1707.04824v1 [math.GT] 16 Jul 2017 This paper is organized as follows. In the Section 2, we introduce the basic concepts from the theory of knot quandles that will be needed in the rest of the paper. Subsection 2.1 contains the definition of the quandle, the augmented quandle, the associated group and some basics about the quandle homomorphisms. In the Subsection 2.2 we define the quandle presentations. In the Subsection 2.3, we define the fundamental quandle of a knot in S3, note some of its properties and describe its presentation. The Section 3 contains the core results of the paper. Recalling the basics of knot symmetries, we first establish how and when a knot symmetry induces a knot quandle (anti)automorphism. Then we use a Theorem of Matveev to show that

Date: January 3, 2018. 2010 Mathematics Subject Classiﬁcation. 57M27, 57M05 (primary), 57M25 (secondary). Key words and phrases. knots, knot symmetries, fundamental quandle.

1 each knot quandle (anti)automorphism is induced by a knot symmetry. In the ﬁnal Section 4, we do some calculations displaying the use of our results in investigating the symmetries of knots. We provide a Phyton code for calculating the Q-group of symmetries from an alternating planar diagram of a knot.

2. Preliminaries 2.1. The definition of a quandle. Definition 2.1. [4] A quandle is an algebraic structure comprising a nonempty set Q with two binary operations (x, y) 7→ x B y and (x, y) 7→ x C y, which satisfies three axioms: (1) x B x = x. (2)( x B y) C y = x = (x C y) B y. (3)( x B y) B z = (x B z) B (y B z).

It follows from the second axiom that the map Sy : Q → Q deﬁned by Sy(x) = x B y is a bijection for any y ∈ Q, and by the third axiom it is an automorphism of Q. By the ﬁrst

quandle axiom the automorphism Sy ﬁxes y. Sy is called an inner automorphism of Q. The subgroup Inn(Q) of all the inner automorphisms is a normal subgroup of Aut(Q) [4].

−1 Example 2.2. Let G be a group. Deﬁne the operations B, C: G × G → G by a B b = b ab −1 and a C b = bab for any a, b ∈ G. It is easy to see that these operations satisfy all the quandle axioms, and the resulting quandle is denoted by Gconj. Thus we obtain a quandle from any group by forgetting multiplication and setting both conjugations as the operations.

Example 2.3. For any quandle Q = (S, B, C), the triple (S, C, B) defines another quandle, which we denote by Qd and is called the dual quandle of Q. Definition 2.4. Let G be a group, acting on itself by conjugation as gh := h−1gh for any g, h ∈ G. An augmented quandle (X,G) is a set X with an action by the group G, written as (x, g) 7→ xg and a function ∂ : X → G, satisfying the augmentation conditions (1) x∂x = x for all x ∈ X, (2) ∂(xg) = g−1(∂x)g for all x ∈ X, g ∈ G. ∂y (∂y)−1 The quandle operations on X are then defined by x B y := x and x C y := x .

Deﬁnition 2.5. Let Q1 and Q2 be quandles. A quandle homomorphism from Q1 to Q2 is a map f : Q1 → Q2 satisfying f(x B y) = f(x) B f(y) for any x, y ∈ Q1. A quandle antihomomorphism from Q1 to Q2 is a map g : Q1 → Q2 satisfying g(xBy) = g(x) C g(y) for any x, y ∈ Q1.

Remark 2.6. Observe that a quandle antihomomorphism g : Q1 → Q2 is actually a quandle d homomorphism from Q1 to Q2.

Lemma 2.7. If f : Q1 → Q2 is a quandle homomorphism, then f(x C y) = f(x) C f(y). If f : Q1 → Q2 is a quandle antihomomorphism, then f(x C y) = f(x) B f(y).

2 Proof. Let f : Q1 → Q2 be a quandle homomorphism and choose x, y ∈ Q1. Denoting w = xCy, we compute w B y = x ⇒ f(w) B f(y) = f(x) ⇒ f(x C y) = f(w) = f(x) C f(y). A similar proof settles the case when f is a quandle antihomomorphism. Lemma 2.8. For any quandle Q, the set AutC (Q) = {f : Q → Q| f a bijection, f a homomorphism or an antihomomorphism} with the composition operation forms a group. Proof. The set Sym(Q) of all bijective maps of the set Q to itself forms a group for composition. We would like to show that AutC (Q) ⊂ Sym(Q) is a subgroup. Thus, we need to show that AutC (Q) is closed under composition and invertation. A composition of two quan- C dle automorphisms is obviously a quandle automorphism. Now let g1, g2 ∈ Aut (Q) be two antihomomorphisms, and choose any x, y ∈ Q. We may compute:

x B y = z ⇒ g2(x) C g2(y) = g2(z) ⇒ g2(z) B g2(y) = g2(x) ⇒ g1(g2(z)) C g1(g2(y)) = g1(g2(x)) ⇒ g1(g2(x)) B g1(g2(y)) = g1(g2(z)) , thus g1 ◦ g2 is a quandle automorphism. In a similar way we show that a composition of a quandle homomorphism and a quandle antihomomorphism (or vice versa) is a quandle antihomomorphism. The inverse of a quandle automorphism is a quandle automorphism. The inverse of any quandle antihomomorphism g is a quandle antihomomorphism, since their composition is the identity automorphism:

x B y = z ⇒ g(x) C g(y) = g(z) ⇒ g(z) B g(y) = g(x) ⇒ −1 −1 −1 −1 g (g(z) B g(y)) = x = z C y ⇒ g (g(z) B g(y)) = g (g(z)) C g (g(y)) . Deﬁnition 2.9. Let Q be a quandle. The associated group As(Q) of the quandle Q is deﬁned as As(Q) = F (Q)/K, where F (Q) is the free group, generated by the set Q, and K is the normal −1 −1 −1 −1 subgroup of F (Q), given by K = {(a B b)b a b, (a C b)ba b | a, b ∈ Q}. Lemma 2.10. Let Q be a quandle and let η : Q → As(Q) be the natural map. Let G be a group with the conjugation quandle Gconj and the natural map ∂ : Gconj → G. Given any quandle homomorphism f : Q → Gconj, there exists a unique group homomorphism f# : As(Q) → G such that f# ◦ η = ∂ ◦ f. η Q As(Q)

f f# ∂ Gconj G

Proof. Let As(Q) = F (Q)/K, where K is the normal subgroup from the Deﬁnition 2.9. The quandle homomorphism f induces a map φ: F (Q) → G. By the deﬁnition of a quandle −1 homomorphism, the quandle operations of the Example 2.2 yield φ(aBb) = φ(b) φ(a)φ(b) and

3 −1 φ(a C b) = φ(b)φ(a)φ(b) for every a, b ∈ F (Q), thus the normal subgroup K lies in the kernel of φ. It follows that φ factors through a unique group homomorphism f# : As(Q) → G.

2.2. Quandle presentations. We recall the following from [7]. Let A be a set we call the alphabet, whose elements we call the letters.A word in A is any ﬁnite sequence, consisting of letters and the symbols (, ), B and C. Deﬁne inductively the set D(A) of admissible words according to the rules: (1) For any a ∈ A, the word a is admissible, (2) If w and z are admissible words, then w B z and w C z are also admissible, (3) There are no other admissible words except those obtained inductively by the rules (1) and (2).

Let R be a set of relations; these are identities of the form ri = si, where ri, si ∈ D(A). Deﬁne an equivalence relation ∼ on the set D(A) as follows: w1 ∼ w2 if and only if w1 can be transformed into w2 by some sequence of the rules (1) - (5) described below: (1) x B x ⇔ x; (2)( x B y) C y ⇔ x; (3)( x C y) B y ⇔ x; (4)( x B y) B z ⇔ (x B z) B (y B z); (5) ri ⇔ si.

The set of equivalence classes D(A)/∼ is denoted by ΓhA|Ri. It is easy to show that ΓhA|Ri is a quandle, and we call hA|Ri the presentation of this quandle.

2.3. The fundamental quandle of a knot. We briefly summarize the following from [8] for the reader’s convenience. Let K be a knot in S3 with a fixed orientation of its normal bundle. 3 Denote by NK the regular neighborhood of K and let EK = S − Int(NK ). Choose a basepoint 0 0 zK ∈ EK , a point zK ∈ ∂NK and a path sK ⊂ EK from zK to zK . Denote by GK := π1(EK , zK ) 0 the fundamental group of the knot. The path sK defines an inclusion of π1(∂NK , zK ) into GK by the formula ba 7→ [sK · a · sK ], where [n] = nb. The image of this inclusion is the peripheral subgroup HK ≤ GK . Denote

ΓK = {homotopy classes of paths in EK from a point in ∂NK to zK } .

During the homotopy the initial point may move around on ∂NK , while the ﬁnal point is kept g ﬁxed. The group GK acts on the set ΓK by bab := [a · g]. Using this action, the set ΓK may be equipped with a structure of an augmented quandle. Any p ∈ ∂NK lies on a unique meridian circle of the normal circle bundle and we denote by mp the loop based at p which follows around the meridian in the positive direction. The image of

mz0 in HK is denoted by mK and is called a meridian of K.

Definition 2.11. The fundamental quandle Q(K) of the knot K is the augmented quandle (ΓK ,GK ) as above with the function ∂ :ΓK → GK defined as follows. Given two classes ba,b ∈ ΓK , which are represented by the paths a and b respectively, define ∂(b) = [b · mb(0) · b], where ·

4 denotes concatenation of paths. This produces the quandle operations

∂(b) ∂(b)−1 ba B b = ba = [a · b · mb(0) · b] and ba C b = ba = [a · b · mb(0) · b] Remark 2.12. For a knot K with a ﬁxed orientation of its normal bundle, denote by Kd the same knot with the opposite orientation of the normal bundle. It follows that Q(Kd) is the dual quandle of Q(K), as deﬁned in the Example 2.3.

Lemma 2.13. The function ∂ :ΓK → GK is a bijection.

Proof. Let ∂ba = ∂b for two elements ba,b ∈ ΓK . It follows that there exists a homotopy ht : [0, 1] → EK , such that h0 = ama(0)a and h1 = bmb(0)b. Using a suitable reparametrization, we may assume that h| 2 is a homotopy between the paths a and b, thus a = b. [ 3 ,1] b To see that ∂ is surjective, observe that the elements ∂x for x ∈ ΓK are exactly the generators of GK in the Wirtinger presentation of the knot group.

Lemma 2.14. [8, Lemma 2] For any knot K, the action of the group GK on Q(K) is transitive. The stabilizer subgroup of the element sbK = [sK ] ∈ Q(K) coincides with HK . Proof. Choose two paths a and b which represent the homotopy classes ba,b ∈ Q(K). Choose a g path n ⊂ ∂NK from a(0) to b(0) and let gb = [a·n·b]. Then we have bab = [a·a·n·b] = [n·b] = b. By the definition of HK , each element bh ∈ HK has the form bh = [sK · n · sK ] for some loop 0 bh n in ∂NK starting and ending at zK . Thus, sbK = sbK and HK acts trivially on sbK . On the gb other hand, if sbK = sbK for some gb ∈ GK , then there is a homotopy H(t, u) between the paths H(0, u) = sK · g and H(1, u) = sK . Let H(t, 0) = n ⊂ ∂NK be the path, traced by the starting point of the path sK · g under this homotopy. Then the path sK · n · sK · g is homotopic to a constant, which means that gb = [sK · n · sK ] lies in HK . Example 2.15. The presentation of the fundamental quandle of a knot. Let K be a knot, given by a knot diagram DK . Label the arcs of DK by y1, . . . , yn. Each crossing of the diagram consists of an overcrossing arc yj and two arcs yi and yk of the undercrossing strand. If we see the arc yi on the right (and yk on the left) of yj when passing along yj, then we define the crossing relation yi B yj = yk. Define a quandle

Γhy1, . . . , yn| crossing relations of DK i .

It can be shown that Γhy1, . . . , yn| crossing relations of DK i is isomorphic to the fundamental quandle Q(K) [7]. Here we oﬀer a brief discussion of this correspondence.

For each arc yi of the diagram DK , choose a path xi from a point on ∂NK , corresponding to the arc yi, to the basepoint. Let xi be such that wherever its projection intersects the diagram DK , it goes over the knot K. The homotopy classes of the paths x1, . . . , xn represent the elements of the set Γ . The augmentation map ∂ :Γ → G is given by ∂(x ) = [x · m · x ] K K K bi i xi(0) i and the fundamental group GK is actually generated by the images ∂(xb1), . . . , ∂(xbn). Consider a crossing of DK with the overcrossing arc yj and the undercrossing arcs yi and yk. If we see the arc yi on the right (and yk on the left) of yj when passing along yj, then we may see the homotopy between xi ·(xj ·mxj (0) ·xj) and xk in the Figure 1. This homotopy implies the crossing relation xbi B xbj = xbk in the fundamental quandle Q(K).

5 yi

yk ∂(xj)

xk ∗

Figure 1. The illustration of the crossing relation

Lemma 2.16. The fundamental group GK is the associated group As(Q(K)) of the fundamental quandle of the knot K.

Proof. Recall the Wirtinger presentation of the fundamental group of a knot. Using notation

from the Example 2.15, the fundamental group GK is generated by the homotopy classes ∂(xb1), . . . , ∂(xbn). Therefore, the augmentation map ∂ :ΓK → GK is a bijection between the generating sets of Q(K) and GK . In the associated group As(Q(K)), every crossing relation −1 xbi B xbj = xbk is equivalent to the relation xbi = xbjxbkxbj , which yields the Wirtinger relation −1 ∂(xbi) = ∂(xbj)∂(xbk)∂(xbj) of this crossing in the fundamental group GK . Remark 2.17. Let F : Q(K) → Q(K) be a quandle automorphism of the fundamental quandle. We may specify how F interfers with the action of GK , deﬁned above. Let ba,b be two elements of Q(K), represented by the paths a and b. Let gb ∈ GK be an element of the fundamental group. By the Lemma 2.16, the element gb may be represented as gb = ∂bc for some bc ∈ Q(K). By the Lemma 2.10, the automorphism F induces a unique group isomorphism F# : GK → GK , for which ∂F (bc) = F#(gb). Then we have

g ∂c ∂F (c) F#(g) F (bab) = F (ba b) = F (ba B bc) = F (ba) B F (bc) = F (ba) b = F (ba) b .

3. Knot symmetries and the automorphisms of the knot quandle Recall that the symmetry group of a knot K in S3 is the mapping class group of the pair (S3,K), meaning the fourth term in the short exact sequence

3 3 3 1 → Aut0(S ,K) → Aut(S ,K) → MCG(S ,K) → 1 ,

3 3 3 where Aut(S ,K) denotes the group of homeomorphisms of the pair (S ,K) and Aut0(S ,K) denotes the normal subgroup of those homeomorphisms which are isotopic to the identity. According to Hoste et al. [3], there are four types of homeomorphisms in Aut(S3,K): (1) those which preserve the orientations of K and of S3, (2) those which reverse the orientation of K and preserve the orientation of S3, (3) those which preserve the orientation of K and reverse the orientation of S3, (4) those which reverse the orientation of K and of S3.

6 For an oriented knot K in S3, denote by rK the same knot with the opposite orientation, by mK the mirror image of K and by rmK the mirror image of K with the opposite orientation. class symmetries knot symmetries knot equivalences c (1) chiral, noninvertible + (1), (3) + amphichiral, noninvertible K = mK - (1), (4) - amphichiral, noninvertible K = rmK i (1), (2) chiral, invertible K = rK a (1), (2), (3), (4) + and - amphichiral, invertible K = rK = mK = rmK

3 3 Proposition 3.1. Let f :(S ,K) → (S ,K) be a homeomorphism for which [f|∂NK ] = ±1 ∈ 2 MCG(T ). Then f induces a map f∗ : Q(K) → Q(K) that is either a quandle automorphism or a quandle antiautomorphism.

3 3 Proof. Let f :(S , K, zK ) → (S , K, zK ) be a homeomorphism. The underlying set ΓK of the fundamental quandle Q(K) is actually the relative homotopy group π1(EK , ∂NK ∪ {zK }, zK ). ∼ ∼ Since f(K) = K, it follows that f(EK ) = EK and f(∂NK ) = ∂NK , thus f induces a map ∼ f∗ :ΓK = π1(EK , ∂NK ∪ {zK }, zK ) → π1(f(EK ), f(∂NK ) ∪ {f(zK )}, f(zK )) = ΓK 0 on the relative homotopy group. Define f∗(ba) = [f ◦ a], where ba = [a]. If a and a are two different representatives of the class ba ∈ ΓK , then there exists a homotopy ht : [0, 1] → EK such 0 0 that h0 = a and h1 = a . Then f ◦ ht is a homotopy from f∗(ba) to f∗(ab), thus f∗ is a well defined map on ΓK . Let ba,b ∈ ΓK be represented by the respective paths a and b. The quandle operation on Q(K) is defined by ba B b = [a · b · mb(0) · b], thus we have f∗(ba B b) = [f(a · b · mb(0) · b)] = [f(a) · f(b) · f(mb(0)) · f(b)] .

Now f(mb(0)) is a loop in f(∂NK ) based at f(b(0)). Since the restriction f|∂NK is isotopic to ± identity, the loop f(mb(0)) also represents a meridian circle of K, which may be either mf(b(0)) or mf(b(0)). It follows that ( f∗(ba) B f∗(b) if f preserves the orientation of the normal bundle, f∗(ba B b) = f∗(ba) C f∗(b) if f reverses the orientation of the normal bundle, therefore the induced map f∗ : Q(K) → Q(K) is a quandle (anti)homomorphism. Since f is surjective, the induced map f∗ is a surjective homomorphism on Q(K). To show the injectivity of f∗, let f∗(ba) = f∗(b) for two elements ba,b ∈ Q(K). Choosing representatives a and b of the respective classes ba and b, there exists a homotopy ht : [0, 1] → EK for which −1 h0 = f ◦a and h1 = f ◦b. Then gt = f ◦ht is a homotopy from a to b, thus ba = b ∈ Q(K).

Example 3.2. Consider the knot 51, given by the diagram of the Figure 2. It is known that 3 3 the knot 51 is 5-periodic, thus there exists a nontrivial homeomorphism f :(S , 51) → (S , 51) of order 5. Let us ﬁnd the corresponding quandle automorphism. The fundamental quandle has a presentation:

Q(51) = ha, b, c, d, e| b B a = e, c B b = a, d B c = b, e B d = c, a B e = di .

7 Consider the map F : Q(51) → Q(51), given by F (a) = b, F (b) = c, F (c) = d, F (d) = e and F (e) = a. Since F preserves the crossing relations of the above presentation, it deﬁnes a quandle automorphism of order 5.

b d

a e

Figure 2. The knot 51

By the Proposition 3.1, there is a subgroup of the automorphism group Aut(S3,K) which acts on the fundamental quandle Q(K) by quandle (anti)automorphisms. Every knot symmetry that preserves the peripheral structure induces a quandle (anti)automorphism. In the following, we offer some kind of a reverse to this correspondence. We need the important result of Waldhausen [10]. For the definition of an incompressible surface, we refer the reader to [10, page 58]. Definition 3.3. Let M be a compact connected 3-manifold. M is called irreducible if any smooth submanifold S ⊂ M homeomorphic to a sphere bounds a subset D which is homeomorphic to the closed 3-ball. An irreducible manifold M, which is not a 3-ball, is called sufficiently large if it contains an incompressible surface. M is called boundary irreducible if its boundary ∂M is incompressible. Theorem 3.4 (Waldhausen, Corollary 6.5 [10]). Suppose that M and N are irreducible and

boundary irreducible. Let M be suﬃciently large and let ψ : π1(N) → π1(M) be an isomorphism preserving the peripheral structure. Then there exists a homeomorphism f : N → M inducing ψ. In his pioneering work on quandles (which he called distributive grupoids), Sergei Matveev proved the fact that the knot quandle is a complete knot invariant. Both our Theorem ?? and its proof are actually an adapted form of the Matveev’s theorem [8, Theorem 2].

Theorem 3.5. Let F : Q(K1) → Q(K2) be an isomorphism of the quandles of two nontrivial 3 3 knots K1 and K2. Then there exists a homeomorphism f :(S ,K1) → (S ,K2) and some g g ∈ GK2 such that f∗ = F . The homeomorphism f preserves the orientation of the normal bundle of K1.

3 Proof. Denote by EKi = S − Int(NKi ) the complement of a regular neighbourhood of Ki. The

manifold EKi is irreducible and suﬃciently large. Since Ki is nontrivial, EKi is also boundary

8 irreducible. Remember that z ∈ E , z0 ∈ ∂N are the chosen basepoints and s is Ki Ki Ki Ki Ki the chosen path in E from z0 to z . The action of G on Q(K ) is transitive by the Ki Ki Ki K2 2 Lemma 2.14; thus there exists a g ∈ G such that F (s )g = s . Deﬁne an isomorphism K2 bK1 bK2 F : Q(K ) → Q(K ) by F (a) = F (a)g and it follows that F (s ) = s . By the Lemma b 1 2 b b bK1 bK2

2.16, the group GKi is associated to Q(Ki) . Therefore, the isomorphism Fb : Q(K1) → Q(K2) induces a unique isomorphism φ: GK1 → GK2 by the Lemma 2.10. By the Lemma 2.14, the peripheral subgroup H ≤ G equals the stabilizer subgroup of the element s , which is Ki Ki bKi the same as the centralizer subgroup of ∂s = m . Since F (s ) = s , the restriction bKi Ki b bK1 bK2 φ| is an isomorphism of H to H and φ(m ) = m . Thus φ: G → G is an HK1 K1 K2 K1 K2 K1 K2 isomorphism, preserving the peripheral structure, and by the Waldhausen’s theorem 3.4 there exists a homeomorphism f : EK1 → EK2 inducing φ. Since φ(mK1 ) = mK2 , the map f extends 3 3 to a homeomorphism f :(S ,K1) → (S ,K2) which preserves the orientation of the normal bundle. We have

∂Fb(ba) = φ(∂ba) = [f ◦ ∂a] = [f ◦ a ◦ ma(0) ◦ a] = [f(a) ◦ mf(a(o)) ◦ f(a)] = ∂[f ◦ a] = ∂f∗(ba) , and by the Lemma 2.13 it follows that Fb(ba) = f∗(ba) for any ba ∈ Q(K1). The map f induces g the quandle isomorphism Fb : Q(K1) → Q(K2), thus f∗ = Fb = F .

Corollary 3.6. Let K1 and K2 be two nontrivial knots with isomorphic fundamental quandles. Then either K1 and K2 are equivalent knots, or the knot K1 is equivalent to rmK2. 3 3 Proof. By the Theorem 3.5, there exists a homeomorphism f :(S ,K1) → (S ,K2) which fixes the orientation of the normal bundle. The orientation of a knot, coupled with the orientation of its normal bundle, gives an orientation of the ambient manifold S3. It follows that if f 3 preserves the orientation of the knot, it must also preserve the orientation of S and thus K1 is equivalent to K2. If, on the other hand, f reverses the orientation of the knot, it must also 3 reverse the orientation of S , and thus K1 is equivalent to rmK2. Proposition 3.7. If there exists an antiautomorphism of the fundamental quandle of a nontrivial knot K, then either K = rK or K = mK. Proof. Let F : Q(K) → Q(K) be an antiautomorphism. By the Remarks 2.6 and 2.12, F defines an isomorphism between the fundamental quandles Q(K) and Q(Kd). It follows by the Theorem 3.5 that there exists a homeomorphism f :(S3,K) → (S3,Kd) that takes the orientation of the normal bundle of K to the orientation of the normal bundle of Kd (which is opposite to that of K). If f preserves the orientation of S3, then it must reverse the orientation of K and thus K = rK. If, on the other hand, f reverses the orientation of S3, then it must preserve the orientation of K and thus K = mK. Remark 3.8. It follows from the Corollary 3.6 and the Proposition 3.7 that we cannot retrieve the information about the knot’s orientation from its fundamental quandle. The knot quandle contains only the information about the orientation of the normal bundle of K, which defines the quandle operations. This is why we will use the term symmetry of a knot for any homeomorphism (S3,K) → (S3,K), often without knowing (or specifying) whether or not it preserves the orientation of K or its ambient manifold S3.

9 Corollary 3.9. Let F : Q(K1) → Q(K2) be an isomorphism of the quandles of two nontrivial knots K and K , for which F (s ) = s . Then there exists a homeomorphism f :(S3,K ) → 1 2 bK1 bK2 1 3 (S ,K2), preserving the orientation of the normal bundle, such that f∗ = F .

Proof. If F (s ) = s , then following the proof of the Theorem 3.5 we may take g = 1 and bK1 bK2 thus Fb = F = f∗. Corollary 3.10. Let K be a nontrivial knot. For any element h of the peripheral subgroup

HK , the inner automorphism Sh is induced by a homeomorphism.

Proof. By the Lemma 2.14, HK is the stabilizer subgroup of the element sbK . The statement then follows from the Corollary 3.9. Proposition 3.11. Let F : Q(K) → Q(K) be an (anti)automorphism of the fundamental quandle of a nontrivial knot K. Then F is induced by a homeomorphism that preserves the orientation of the normal bundle if F is an automorphism, and reverses the orientation of the normal bundle if F is an antiautomorphism.

Proof. Let K = (K, o, s ) denote the knot K with a ﬁxed orientation of the normal bundle, 1 bK1 equipped with a ﬁxed element s . Let s ⊂ E be a path that represents the element bK1 K2 K F (s ) ∈ Q(K). Let K = (K, ±o, s ) denote the knot K with the same orientation of the bK1 2 bK2 normal bundle if F is an automorphism, and the reverse orientation if F is an antiautomorphism, equipped with the element s = F (s ). Now F : K → K is a quandle isomorphism for bK2 bK1 1 2 which F (s ) = s . By the Corollary 3.9, there exists a homeomorphism f :(S3,K) → bK1 bK2 3 (S ,K) such that f∗ = F . The homeomorphism f preserves the orientation of the normal bundle if F is an automorphism, and reverses the orientation of the normal bundle if F is an antiautomorphism.

4. Examples and calculations

a e

g i b

c f d h

Figure 3. The knot 940

10 Example 4.1. Consider the knot 940 in the Rolfsen knot table, whose diagram is given on the Figure 3. The presentation of its fundamental quandle is given by

Q(940) = ha, b, c, d, e, f, g, h, i| i B e = a, h B a = g, b B d = a, f B b = g, c B i = b, f B c = e, c B h = d, e B g = d, i B f = hi .

Deﬁne a map F : Q(940) → Q(940) by F (a) = g, F (b) = h, F (c) = i, F (d) = a, F (e) = b, F (f) = c, F (g) = d, F (h) = e, F (i) = f. It is easy to check that F preserves the crossing relations of the above presentation and thus deﬁnes a quandle automorphism of order 3. It follows by the 3 3 Proposition 3.11 that there exists a homeomorphism f :(S , 940) → (S , 940) of order 3.

a c

Figure 4. The trefoil knot 31

Example 4.2. The trefoil knot 31 has the following presentation of the fundamental quandle, obtained from the diagram in the Figure 4:

Q(31) = ha, b, c| a B b = c, b B c = a, c B a = bi .

The corresponding presentation of the fundamental group GK is given by

−1 −1 −1 GK = h∂a, ∂b, ∂c| (∂b) (∂a)(∂b) = ∂c, (∂c) (∂b)(∂c) = ∂a, (∂a) (∂c)(∂a) = ∂bi .

g Consider the element g = (∂a)(∂b) = (∂b)(∂c) = (∂c)(∂a) ∈ GK . We calculate a = a B b B c = g g c, b = b B c B a = a and c = c B c B a = b, so the inner automorphism Sg is a quandle automorphism of order 3. It follows by the Proposition 3.11 that the trefoil knot admits a symmetry of order 3.

Example 4.3. The ﬁgure-eight knot 41 has the following presentation of the fundamental quandle, obtained from the diagram in the Figure 5:

Q(41) = ha, b, c, d| c B a = b, a B d = b, a B c = d, c B b = di . Define a map F : Q(K) → Q(K) by F (a) = c, F (b) = d, F (c) = a and F (d) = b. It is easy to see that F preserves the crossing relations in the above presentation of Q(41) and thus defines a quandle automorphism of order 2. It follows by the Proposition 3.11 that the figure-eight knot admits a symmetry of order 2.

11 c

a b

Figure 5. The ﬁgure-eight knot 41

By the Proposition 3.11, each (anti)automorphism of the fundamental quandle of a nontrivial knot is induced by a knot symmetry of K. Moreover, the existence of a quandle automorphism of order p implies that K admits a symmetry of order p. Thus, by observing the set of quandle (anti)automorphisms of Q(K) we might learn something about the knot symmetries of K. Of course, the set AutC (Q(K)) of all quandle automorphisms is usually too large to grasp in its entirety. Instead, we might try to consider a simple subset of AutC (Q(K)) that is directly related to a given diagram of K.

Definition 4.4. Let DK be a diagram of a knot K with an ordered set of arcs (x1, . . . , xn), which defines a presentation hx1, . . . , xn| crossing relations i of the fundamental quandle Q(K). A permutation σ ∈ Sn is called a Q-permutation for DK if the map xi 7→ xσ(i) defines a quandle (anti)automorphism of Q(K). The Q-group of the diagram DK is the subgroup of Sn containing all the Q-permutations of DK .

If σ ∈ Sn is a Q-permutation, then the map Fσ : Q(K) → Q(K), defined on the generating set by Fσ(xi) = xσ(i), defines a quandle (anti)automorphism. By the Proposition 3.11, there exists 3 3 a homeomorphism f :(S ,K) → (S ,K) such that f∗ = Fσ. Therefore, every Q-permutation defines a knot symmetry of K, and the Q-group of DK corresponds to some subset of the automorphism set Aut(S3,K).

Lemma 4.5. If DK is a knot diagram with n arcs, then the order of the Q-group of DK is at most 2n.

Proof. Let (x1, . . . , xn) be the ordered set of arcs of the diagram DK . Pick a crossing relation xi B xj = xk. If σ is a Q-permutation, then the image σ(i) and the information about whether Fσ deﬁnes an automorphism or an antiautomorphism of Q(K) determine the images σ(j) and σ(k) and the images of all the other integers are determined by the crossing relations. A good thing about the Q-group is that it may be quite easily calculated from the planar representation of a knot diagram DK . By the Lemma 4.5, the order of the Q-group is bounded by twice the size of the knot diagram, so the time complexity of a suitable computer imple- mentation should not be a problem. In the Subsection 4.1 we provide a Phyton code for the calculation of the Q-group from an alternating planar diagram.

12 The following table shows the calculations of the Q-groups of the minimal diagrams of some alternating knots. We used the diagrams given in the Rolfsen knot table at Knotatlas [6].

Knot K Q-group of DK 31 D3 41 Z4 51 D5 52 Z2 61 Z2 62 {id} 63 {id} 71 D7 72 Z2 73 Z2 74 Z2 940 Z6

4.1. The Phyton code. from sympy.combinatorics.permutations import Permutation from sympy.combinatorics.generators import symmetric Permutation.print_cyclic=False def quandlesym(PD): ### calculates the $Q$-group of an alternating knot diagram### L,P1,R=[],[],[] P=list(symmetric(len(PD))) for X in PD: L=L+[X[2],X[4]] for X in PD: if X[0] == 1: P1.append([L.index(X[1])//2,L.index(X[2])//2,L.index(X[3])//2]) else: P1.append([L.index(X[3])//2,L.index(X[2])//2,L.index(X[1])//2]) for i in range(1,len(P)+1): v1,v2 = True,True for Y in P1: if [P[i-1](Y[0]),P[i-1](Y[1]),P[i-1](Y[2])] not in P1: v1 = False if [P[i-1](Y[2]),P[i-1](Y[1]),P[i-1](Y[0])] not in P1: v2 = False if v1 or v2: R.append(P[i-1]) return(R)

13 References [1] R. Elliot, Alexander Polynomials of Periodic Knots: A Homological Proof and Twisted Extension. Princeton Undergraduate Thesis, 2008. [2] R. Fenn, C. Rourke, Racks and links in codimension two. J. Knot Theory Ramifications, 1 (1992), 343–406. [3] J. Hoste, M. Thistlethwaite, J. Weeks, The first 1701936 knots. Math. Intell. 20, 33–48, 1998. [4] J. Joyce, An algebraic approach to symmetry with applications to knot theory. A dissertation, 1979. [5] A. Kawauchi, A survey of knot theory. Springer Science & Business Media, 1996. [6] The Knot Atlas: The Rolfsen Knot Table., http://katlas.org/wiki/The_Rolfsen_Knot_Table. [7] V. Manturov, Knot theory. CRC Press, 2004. [8] S. V. Matveev, Distributive grupoids in knot theory. Math. USSR Sbornik, 47(1):,73–83, 1984. [9] M. Elhamdadi, S. Nelson, Quandles – An introduction to the algebra of knots. volume 74 of Student Mathe- matical Library. American Mathematical Society, Providence, RI, 2015. [10] F. Waldhausen, On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2) 87: 56–88, 1968.

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